Gentle region (extended version)

This is an extended version of the Gentle region article.

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Margo Schulter, in a tuning list posting, defined the "gentle region" of temperaments with a fifth as generator as that of fifths about 1.49 to 2.65 cents sharp; later amending that to from 1.49 to 3.04 cents sharp. We can consider the first region to extend from fifths of size 17\29 to 64\109, and the extended region to reach 47\80. If we remove the restriction to tempering based on chains of fifths, we find that notable equal divisions in the smaller gentle region include multiples of 29edo, 46edo, 75edo, 104edo, 109edo, 121edo, 133edo, 155edo, 162edo, 167edo, 179edo, 191edo, 201edo, 213edo, 225edo and 237edo, plus 63edo and 80edo in the extended region.

Generator		Cents	2-3-7(b)-11-13(b)	Half 8/7+ 1\3 7/61\3 8/7+ Half 7/6		8/7+7/6
(7+10)\29		289.655+413.793	< 29 46 81 100 107|	2\29+3\29 82.759+124.138		6\29+6\29 248.276+248.276
(25+36)\104		288.4615+415.385	< 104 165 292 360 385|	10\104+23\312 115.385+88.4615 20\312+23\208 76.923+132.692		20\104+23\104 230.769+265.385
(18+26)\75		288+416	< 75 119 210~211 259 277|	7\75+17\225 112+90.667 14\225+17\150 74.667+136		14\75+17\75 224+272
	(47+68)\196	287.755+416.3265	< 196 311 549-551 678 725|	37\392+44\588 113.265+89.796 37/588+22/196 75.51+134.694		37/196+44/196 226.531+269.388
		287.713+416.382	< 29 46 81 100 107|+< 46 73 129 159 170|φ	119.283+85.7795 79.522+128.769		238.566+257.3385
	(29+42)\121	287.603+416.529	< 121 192 339~340 419 448|	23\242+9\121 114.05+89.256 23\363+27\242 76.033+133.884		23\121+27\121 228.099+267.769
		287.267+416.978	< 29 46 81 100 107|+< 109 173 306 377 403|φ	116.8205+87.323 78.617+130.984		235.85+261.969
(11+16)\46		286.9565+417.391	< 46 73 129 159 170|	9\92+10\138 117.391+86.9565 3\46+5\46 78.261+130.435		9\46+10\46 234.783+260.87
		286.587+417.884	< 29 46 81 100 107|+<63 100 177 218 233|φ	117.925+88.626 77.635+132.9395		232.936+265.879
	(48+70)\201	286.567+417.91	< 201 319 564 695 703|	39\402+44\603 116.418+87.56 13\201+22\201 77.612+131.343		39\201+44\201 232.836+262.687
	(37+54)\155	286.452+418.0645	< 155 246 435 536 573|	15\155+34\465 116.129+87.742 10\155+17\15 77.419+131.613		30\155+34\155 232.258+263.226
		286.387+418.151	< 46 73 129 159 170|+< 109 173 306 377 403|φ	115.968+87.842 77.312+131.7365		231.935+263.527
	(63+92)\264	286.364+418.182	< 264 419 741 913 976|	51\528+58\792 115.909+87.87 17\264+29\264 77.273+131.818		51\264+58\264 231.818+263.636
(26+38)\109		286.2385+418.349	< 109 173 306 377 403|	21\218+8\109 115.596+88.07 7\109+12\10 77.064+132.11		21\109+24\109 231.192+264.22	Boundary of smaller "gentle region"
	(67+98)\281	286.121+418.505	< 281 446 789 972 1039|	27\281+62\843 115.3025+88.256 18\281+31\28 76.868+132.384		54\281+62\281 230.605+264.769
		286.101+418.533	< 46 73 129 159 170|+< 63 100 177 218 233|φ	116.526+89.264 77.684+133.8965		233.052+267.793
	(41+60)\172	286.0465+418.605	< 172 273 483 595 636|	33\344+38\516 115.116+88.372 11\172+19\17 76.744+132.558		33\172+38\172 230.232+265.116
	(56+82)\235	285.957+418.723	< 235 373 660 813 869|	45\470+52\705 114.894+81.511 15\235+26\23 76.596+132.766		45\235+52\235 229.787+265.532
		285.852+418.864	< 109 173 306 377 403|+< 63 100 177 218 233|φ	114.963+88.4675 76.642+132.701		229.926+265.402
(15+22)\63		285.714+419.048	< 63 100 177 218 233|	6\63+14\189 114.286+88.88 4\63+7\63 76.1905+133.333		12\63+14\63 228.571+266.667
		285.513+419.316	< 46 73 129 159 170|+< 80 127 225 277 296|φ	113.7825+89.2075.855+133.80		227.565+267.606
	(49+72)\206	285.437+419.4175	< 206 327 578~579 713 762|	20\206+15\206 116.505+87.37 40\618+45\41 77.67+131.068		40\206+45\206 233.01+262.136
(34+50)\143		285.315+419.58	< 143 227 401~402 495 529|	14\143+31\429 117.4825+86.71 28\429+31\28 78.322+130.07		28\143+31\143 234.965+260.14
		285.234+419.688	< 63 100 177 218 233|+< 80 127 225 277 296|φ	113.085+89.636 75.39+134.454		226.169+268.909
(54+78)\223		285.202+419.731	< 223 354 626~627 771 825|	43\446+49\669 115.695+87.892 43\669+49\44 77.13+131.839		43\223+49\223 231.39+263.677
(19+28)\80		285+420	< 80 127 225 277 296|	15\160+6\80 112.5+90 5\80+9\80 75+135		15\80+18\80 225+270	Boundary of larger "gentle region"